Electromagnetic Wave

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Maxwell's Equations

\[\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}\]

\[\nabla \cdot \vec{B} = 0\]

\[\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}\]

\[\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}\]

Properties & Key Points:

Maxwell's equations describe the behavior of electric and magnetic fields and their interactions with matter.
\( \vec{E} \): Electric field vector.
\( \vec{B} \): Magnetic field vector.
\( \rho \): Charge density.
\( \vec{J} \): Current density.
\( \epsilon_0 \): Permittivity of free space.
\( \mu_0 \): Permeability of free space.

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Electromagnetic Wave Propagation

\[c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}\]

Properties & Key Points:

Electromagnetic waves consist of oscillating electric and magnetic fields that propagate through space at the speed of light.
\( c \): Speed of light.
\( \mu_0 \): Permeability of free space.
\( \epsilon_0 \): Permittivity of free space.

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Wave Equation for Electromagnetic Waves

\[\frac{\partial^2 \vec{E}}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 \vec{E}}{\partial t^2}\]

\[\frac{\partial^2 \vec{B}}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 \vec{B}}{\partial t^2}\]

Properties & Key Points:

The wave equation describes the propagation of electromagnetic waves in free space.
\( \vec{E} \): Electric field vector.
\( \vec{B} \): Magnetic field vector.
\( c \): Speed of light.
\( x \): Spatial coordinate.
\( t \): Time.

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Energy in Electromagnetic Waves

\[u = \frac{\epsilon_0 E^2}{2} = \frac{B^2}{2\mu_0}\]

Properties & Key Points:

The energy carried by an electromagnetic wave is divided equally between the electric and magnetic fields.
\( u \): Energy density.
\( E \): Electric field strength.
\( B \): Magnetic field strength.
\( \epsilon_0 \): Permittivity of free space.
\( \mu_0 \): Permeability of free space.

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Poynting Vector

\[\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}\]

Properties & Key Points:

The Poynting vector describes the directional energy flux (the energy transfer per unit area per unit time) of an electromagnetic wave.
\( \vec{S} \): Poynting vector (energy flux).
\( \vec{E} \): Electric field vector.
\( \vec{B} \): Magnetic field vector.
\( \mu_0 \): Permeability of free space.

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Electromagnetic Spectrum

\[f = \frac{c}{\lambda}\]

Properties & Key Points:

The electromagnetic spectrum represents the range of all possible frequencies of electromagnetic radiation.
\( f \): Frequency of the wave.
\( c \): Speed of light.
\( \lambda \): Wavelength of the wave.

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Wavelength of Electromagnetic Waves

\[\lambda = \frac{c}{f}\]

Properties & Key Points:

The wavelength of an electromagnetic wave is the distance between two consecutive peaks or troughs.
\( \lambda \): Wavelength of the wave.
\( c \): Speed of light.
\( f \): Frequency of the wave.

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Intensity of Electromagnetic Waves

\[I = \frac{1}{2} \epsilon_0 c E_0^2\]

Properties & Key Points:

The intensity of an electromagnetic wave is the power per unit area carried by the wave.
\( I \): Intensity of the wave.
\( \epsilon_0 \): Permittivity of free space.
\( c \): Speed of light.
\( E_0 \): Maximum electric field strength.

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Energy of a Photon

\[E = h f\]

Properties & Key Points:

The energy of a photon (quantum of electromagnetic radiation) is related to its frequency.
\( E \): Energy of the photon.
\( h \): Planck’s constant.
\( f \): Frequency of the photon.

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Momentum of a Photon

\[p = \frac{E}{c} = \frac{h}{\lambda}\]

Properties & Key Points:

The momentum of a photon is related to its energy and wavelength.
\( p \): Momentum of the photon.
\( E \): Energy of the photon.
\( c \): Speed of light.
\( h \): Planck’s constant.
\( \lambda \): Wavelength of the photon.

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